Find the sum of the infinite geometric series 2 + 2√2+2√2√2+2√2√2√2+..., where each term is obtained by multiplying the previous term by √/2. Full credit will only be given for correct notation, organized work, and showing all steps.
Find the sum of the infinite geometric series 2 + 2√2+2√2√2+2√2√2√2+..., where each term is obtained by multiplying the previous term by √/2. Full credit will only be given for correct notation, organized work, and showing all steps.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Statement
Find the sum of the infinite geometric series \(2 + 2\sqrt{2} + 2\sqrt{2}\sqrt{2} + 2\sqrt{2}\sqrt{2}\sqrt{2} + \ldots\), where each term is obtained by multiplying the previous term by \(\sqrt{2}\).
### Instructions
Full credit will only be given for correct notation, organized work, and showing all steps.
### Solution Explanation
1. **Recognize the pattern**: The given series is a geometric series where each term is formed by multiplying the previous term by \(\sqrt{2}\).
2. **Identify the first term (\( a \)) and common ratio (\( r \))**:
- First term (\( a \)) = 2
- Common ratio (\( r \)) = \(\sqrt{2}\)
3. **Geometric Series Sum Formula**: For an infinite geometric series \(a + ar + ar^2 + ar^3 + \ldots\), the sum \( S \) is given by:
\[
S = \frac{a}{1 - r}
\]
provided that \( |r| < 1 \).
4. **Check the condition**: For the series to converge, \( |r| \) must be less than 1. Since \( \sqrt{2} > 1 \), this series does not converge and thus does not have a finite sum.
**Conclusion**: The infinite geometric series \(2 + 2\sqrt{2} + 2(\sqrt{2})^2 + 2(\sqrt{2})^3 + \ldots\) does not converge because the common ratio \(\sqrt{2}\) is greater than 1. Therefore, the series does not have a finite sum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99289c38-7b50-427e-b7ea-7af0a1fcf022%2Fc8a6a3ee-985e-47fb-96e5-ef1a0113a14a%2F1pa4z35_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the sum of the infinite geometric series \(2 + 2\sqrt{2} + 2\sqrt{2}\sqrt{2} + 2\sqrt{2}\sqrt{2}\sqrt{2} + \ldots\), where each term is obtained by multiplying the previous term by \(\sqrt{2}\).
### Instructions
Full credit will only be given for correct notation, organized work, and showing all steps.
### Solution Explanation
1. **Recognize the pattern**: The given series is a geometric series where each term is formed by multiplying the previous term by \(\sqrt{2}\).
2. **Identify the first term (\( a \)) and common ratio (\( r \))**:
- First term (\( a \)) = 2
- Common ratio (\( r \)) = \(\sqrt{2}\)
3. **Geometric Series Sum Formula**: For an infinite geometric series \(a + ar + ar^2 + ar^3 + \ldots\), the sum \( S \) is given by:
\[
S = \frac{a}{1 - r}
\]
provided that \( |r| < 1 \).
4. **Check the condition**: For the series to converge, \( |r| \) must be less than 1. Since \( \sqrt{2} > 1 \), this series does not converge and thus does not have a finite sum.
**Conclusion**: The infinite geometric series \(2 + 2\sqrt{2} + 2(\sqrt{2})^2 + 2(\sqrt{2})^3 + \ldots\) does not converge because the common ratio \(\sqrt{2}\) is greater than 1. Therefore, the series does not have a finite sum.
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